QALYs Versus WTP. James K. Hammitt

Transcription

1 Risk Analysis, Vol. 22, No. 5, 2002 QALYs Versus WTP James K. Hammitt Quality adjusted life years (QALYs) and willingness to pay (WTP) are alternative measures of the value of reductions in health risk that are often used in evaluating environmental, health, and safety practices. Although both methods are based on individual preferences, the underlying assumptions differ. The different bases yield systematically different conclusions about the relative value of reducing health and mortality risks to individuals that differ in age, preexisting health conditions, income, and other factors. The choice of which method to use depends on judgments about what constraints should be placed on individual preferences and what factors should be considered in aggregating preferences across people. KEY WORDS: Health metric; valuation; quality adjusted life years; willingness to pay 1. INTRODUCTION Comparative risk analysis and life cycle impact assessment attempt to characterize the portfolio of health risks associated with a specific policy or product. To determine whether the net effect of a policy on health is positive or negative, it is often necessary to aggregate disparate health risks, which may be fatal or nonfatal, cause cancer or other disease, and vary in other characteristics. For example, comparative risk analysis of drinking-water-treatment systems requires a method for comparing risks of gastrointesinal illness caused by microbial pathogens with risks of cancer caused by disinfection byproducts. Life cycle assessment of the effects of coal-burning electricity plants requires aggregating fatal and nonfatal injuries in coal mining and transport with respiratory illness and fatalities from particulate air pollution. A number of alternative approaches to valuing changes in health and mortality risks have been developed. Two of the most prominent are the quality adjusted life year (QALY) and willingness to pay (WTP) frameworks. QALYs are used routinely in the medical and public-health fields, whereas Center for Risk Analysis, School of Public Health, 718 Huntington Ave., Boston, MA 02115; tel.: (617) ; fax: (617) ; jkh@hsph.harvard.edu. WTP is widely used in evaluating environmental and transportation-related risks. Although they have been developed in different application areas, the QALY and WTP frameworks share important similarities: both are justified as representing the preferences of individuals, and both are summed across individuals to represent the social value of a change in health risk. However, the specific assumptions underlying the approaches differ in ways that produce systematic differences in the relative values of changes in risks. These differences may lead to different conclusions about whether a policy increases or decreases aggregate health risk. This article provides an introduction to the QALY and WTP approaches, reviews the theoretical foundations underlying them, and examines the implications of differences in the foundations for ranking environmental and other health risks. In particular, it examines the effects of age, health, and income of the affected population on the value of reducing mortality risk. The article is organized as follows. Section 2 reviews the theoretical assumptions of the two approaches. Section 3 examines the implications for valuing current mortality risk and aggregating values of mortality-risk changes across individuals. Section 4 describes empirical methods for estimating values /02/ $22.00/1 C 2002 Society for Risk Analysis

2 986 Hammitt 1 With intervention HRQL Without intervention Fig. 1. QALYs for two hypothetical health profiles. The intervention improves health at all ages and extends longevity from D 1 to D 2. The difference in QALYs is the area between the two health profiles. 0 Age D 1 D 2 under the two approaches. Section 5 discusses how health risks are aggregated across people or combined with other endpoints under the two approaches, and Section 6 concludes. 2. UTILITY-THEORETIC FOUNDATIONS An individual experiences various health states over his or her lifetime. The time path of health states experienced, ending in death, is a health profile. Risks to health and/or longevity may be represented as lotteries (probability distributions) over alternative health profiles, and policies or other interventions that alter health risks alter the probabilities associated with experiencing different health profiles. 1 A utility function is any function that summarizes an individual s preferences, in the sense that it assigns a higher number to a more preferred lottery. Both QALYs and WTP are justified as representing individual utility functions. QALYs assume that preferences over health and longevity depend only on health consequences, and do not depend on other characteristics of the individual or the risk. 2 In contrast, WTP allows for the possibility that preferences over health outcomes depend on individual characteristics, such as 1 Note that a health profile experienced with certainty can be represented as a degenerate lottery that assigns probability one to the certain health profile and probability zero to all other profiles. 2 Technically, preferences over health quality and longevity must be utility independent (1) of other characteristics of the individual and the risk. wealth, as well as on characteristics of the risk, such as whether it is perceived to be uncontrollable, unfamiliar, or dreaded QALYs The QALY framework provides a method for measuring the value of a health profile in terms of the duration of an equally preferred health profile free of any health impairment. The number of QALYs in a specified health profile is calculated as the qualityweighted lifespan: M QALYs = q i T i. (1) In Equation (1), lifespan is divided into M periods that are indexed by i. The periods are defined so that only one health state is experienced in each period. The duration of period i is T i and the health-related quality of life (HRQL) associated with that period is characterized by a weight q i. The value of an intervention that affects health and/or longevity is measured as the difference in QALYs between the health profiles obtained with and without the intervention, as illustrated in Fig. 1. The HRQL is a number that represents the quality of health. 3 It is scaled so that a value of one i=1 3 Several terms, including health-related quality of life, health status, and functional status, are used in the literature to designate a variety of single and multidimensional measures of health.

3 QALYs Versus WTP 987 corresponds to perfect or excellent health, and a value of zero corresponds to health that is equivalent to death (i.e., an individual would not care if he or she were to live the rest of his or her lifespan in such a state or die immediately). Typically, q is between one and zero, but values of q less than zero can be used to represent states of health that are worse than death. The conditions under which QALYs represent a valid individual utility function were identified by Pliskin et al. (4) These authors restrict their attention to the special case of chronic (constant) health states, for which Equation (1) simplifies to: QALYs = q T (2) where T is remaining lifespan and q is the HRQL for the constant health state in which the individual will live until death. In this case, QALYs represent a valid utility function for an individual if his or her preferences satisfy the following conditions. 1. Mutual utility independence. This condition has two parts: (a) preferences between lotteries on health states, holding duration of life constant, do not depend on remaining lifespan; and (b) preferences between lotteries on lifespan, holding health state constant, do not depend on health state. An example of part (a) is if an individual is indifferent between living 40 years in good health and a lottery between living 40 years in excellent health or in fair health, she is also indifferent between living 25 years in good health and a lottery between living 25 years in excellent or fair health. 4 An example of part (b) is if an individual is indifferent between living 30 years and a lottery between living 40 years and 25 years, with all years lived in excellent health, then she is also indifferent between living 30 years and a lottery between living 40 years and 25 years, with all years lived in fair health. Mutual utility independence is necessary for utility to be represented as a product of separate health and longevity terms. (1) I follow the U.S. Public Health Service Panel (2) and Dolan (3) in using the term HRQL to designate the one-dimensional utility value q i. 4 As described below, health states are typically described in much greater detail. Simple descriptions such as excellent, good, and fair are used here for illustration. The notation M-N lottery denotes a lottery where the probability of the first outcome is M% and the probability of the second outcome in N%. 2. Constant proportional tradeoff of longevity for health. The fraction of remaining lifespan the individual would be willing to sacrifice to improve his or her health from one state to another does not depend on his or her remaining lifespan. For example, if an individual is indifferent between living 40 years in fair health and 30 years in excellent health, she is also indifferent between living 20 years in fair health and 15 years in excellent health. This condition implies that the HRQL associated with a health state does not depend on the length of time spent in that state. 3. Risk neutrality over lifespan. Holding health state constant, the individual prefers whichever lottery on longevity provides the greatest life expectancy. For example, the individual would prefer to live 41 years to a lottery between living 50 and 30 years, and she would prefer that lottery to living 39 years (where all years are lived in the same health state, e.g., excellent health). A risk-adjusted form of QALY (which does not require risk neutrality) has also been developed (4) but is rarely used in practice. In the risk-adjusted case, the simple and ethically appealing calculation of changes in social utility as the population sum of individual changes in QALYs is inconsistent with individual preferences. This follows because the value of a health profile to an individual is a nonlinear function of duration, and so the individual s utility is not equal to the sum of his or her quality-weighted life years. Recently, Bleichrodt et al. (5) and Miyamoto et al. (6) proposed alternative and simpler conditions that imply an individual s preferences over lotteries on chronic health profiles can be represented by QALYs. One condition is that the individual is indifferent among all health states when his or her lifespan is zero (the zero condition ). If, in addition, he or she is risk neutral over lifespan for each health state (which implies that longevity is utility independent of health state), then his or her preferences can be described by Equation (2). (5) Alternatively, if his or her preferences for lotteries on lifespan holding health constant do not depend on the health state (i.e., lifespan is utility independent of health), then his or her preferences

4 988 Hammitt can be represented as a form of risk-adjusted QALY. (6)5 For the more general case in which health can vary over the lifespan (Equation (1)), an additional condition is required. 4. Additive independence across periods. The individual s preferences for lotteries on health in any subset of the periods do not depend on health in other periods. (3) For example, if the individual is indifferent between (a) spending 10 years in good health and (b) spending five years in good health followed by a lottery between five years in excellent health and five years in poor health, then she is also indifferent between (c) spending five years in excellent health followed by five years in good health and (d) spending five years in excellent health followed by a lottery between five years in excellent health and five years in poor health. 6 Additive independence also implies that the individual is indifferent between health profiles offering the same total time spent in each health state, regardless of the sequence in which the health states are experienced. This condition is necessary for QALYs to be calculated as the sum of HRQL-weighted time spent in each health state. When QALYs are added across individuals (in order to evaluate social policies), it is generally considered appropriate to discount future QALYs at the same rate at which future monetary costs are discounted. (2) Discounting QALYs is justified as treating individuals equally when resources are allocated using cost-effectiveness ratios: if the costs of an intervention are discounted but the effects (added QALYs) are not, then an intervention can be made to appear more favorable simply by postponing its implementation. (7) Discounting future QALYs con- 5 The form of risk-adjusted QALY consistent with the Pliskin et al. (4) assumptions is more restrictive than the form consistent with the Miyamoto et al. (6) assumptions. The Pliskin et al. (4) assumptions require that risk posture with respect to longevity, holding health constant, satisfy constant relative risk aversion (or constant relative risk proneness). The Miyamoto et al. (6) assumptions impose no constraint on risk posture with respect to longevity (one can be risk averse for some range of lifespan and risk seeking for another), and do not even require that an individual always prefers a longer lifespan. 6 The alternatives (c) and (d) can be obtained from alternatives (a) and (b) by changing the health state for the first five years from good to excellent. flicts with the utility-theoretic justification, although the conflict could be remedied by substituting the present value of duration (using the appropriate discount rate) in Conditions 2 and 3 above. (8) Empirical research suggests that individual preferences for health and longevity often violate the conditions under which QALYs provide a valid utility function for individual health. (4,8 16) These violations are often small and idiosyncratic, and QALYs are considered by many to provide a reasonable starting point for representing preferences. (2,3) Two alternative measures, DALYs and HYEs, are closely related to QALYs. Disability adjusted life years (DALYs) (17,18) are similar to QALYs except that they incorporate a weighting factor that depends on age and measure the loss of longevity and health from an idealized health profile. 7 The age-weighting factor represents a judgment that years lived in young adulthood and middle age contribute more to a society than years lived as a child or in old age. The factor is proportional to y e βy where y is age in years and β is a parameter conventionally set equal to (17) For this value of β, the weighting factor is largest at age 25; it is about three-fourths as large at ages 10 and 50, and half as large at ages 6 and 67. For evaluating changes in health risk, the measurement of health as adeficit from a reference health profile has no effect as the reference health profile cancels. Healthy years equivalent (HYEs) (19) may be viewed as a less restrictive form of QALYs. The HYE for a specified health profile is simply the number of years lived in perfect health that the individual judges to be as desirable as a specified health profile. There is no requirement that HYEs satisfy any of the four assumptions required of QALYs; thus HYEs are much more flexible. Concomitantly, because HYEs impose so little structure on preferences, it is necessary to elicit the HYE directly for each health profile of interest; it cannot be calculated from data on duration and preferences for different health states. For example, because HYEs do not require that constant proportional tradeoff (Assumption 2) is satisfied, one cannot assume that the ratio of HYEs to time spent in a chronic health state is independent of lifespan. The HYE framework admits the possibility that an individual may be indifferent between 40 years in poor health and 20 years in excellent health, and also indifferent between 10 years in poor health and nine years in excellent health. Perhaps because they 7 In contrast, QALYs measure the value of a health profile relative to immediate death.

5 QALYs Versus WTP 989 Wealth Fig. 2. Indifference curves for survival probability and wealth. Starting from A, WTP to improve the probability of surviving the current period from p 0 to p 1 is equal to the distance B C. WTA compensation in place of improving health from p 0 to p 1 is equal to the distance D A. The distance B C also represents WTP to prevent a reduction from p 1 to p 0, and the distance D A represents WTA compensation to permit a reduction from p 1 to p 0. WTA w 0 WTP D A B C 0 p 0 p 1 1 Survival (= 1 risk) impose so little structure on preferences, HYEs have not been widely used in practice WTP The WTP approach reflects conventional microeconomic principles. Anything over which an individual has preferences, including lotteries on health profiles, can be described as an economic good. An individual s preference for one lottery over another can be represented in terms of a change in income or wealth, which can be used to purchase other goods. There are two alternative measures of an individual s willingness to trade money and health: WTP and willingness to accept (WTA). Consider the value to an individual with wealth w 0 of moving from health profile H 0 to a preferred health profile H 1. Her utility is a function of the health profile and wealth, u(h, w). The value of the improvement may be measured as: 1. WTP for improvement (compensating variation), the value of c 0 satisfying u(h 0,w 0 ) = u(h 1, w 0 c 0 ). The name implies that the loss of wealth c 0 compensates for the gain in health, leaving the individual no better or worse off than without the health improvement. 2. WTA in place of improvement (equivalent variation), the value of e 0 satisfying u(h 0, w 0 + e 0 ) = u(h 1, w 0 ). The payment is equivalent to the health gain, in that the individual is equally well off whether she obtains the payment or the health improvement. 8 Fig. 2 illustrates WTP and WTA for changes in current-period mortality risk, holding the lottery on health and survival in future time periods constant. The figure illustrates two indifference curves for the probability of surviving a specified period (e.g., the current year) and wealth available for spending on other goods. An indifference curve is defined as a set of points such that the individual judges all points along it to be equally desirable. Points above and to the right of the indifference curve are preferred, as they represent larger survival probability and/or greater wealth. Under plausible assumptions (described in Section 3.2), the indifference curves relating survival probability and income are downward sloping and convex, as illustrated. The initial position with survival probability p 0 and wealth w 0 is labeled A. An increase in survival probability to p 1 would shift the individual to B, on a higher indifference curve. The individual s WTP for 8 Note that WTA to forego an improvement from H 0 to H 1 is different than willingness to accept compensation for a reduction in health from H 0 to some less desired health profile. One can also define WTP to prevent a reduction from H 1 to H 0 and WTA to permit a reduction from H 1 to H 0 (see Fig. 2).

6 990 Hammitt this increase in survival probability is given by the vertical distance between the two indifference curves at p 1,B C. Alternatively, the individual could achieve the same increase in utility by moving to point D, which involves no change in his or her survival probability but an increase in his or her wealth. The individual s WTA compensation in lieu of the survival improvement is given by the vertical distance between the two indifference curves at p 0,D A. If the risk reduction p 1 p 0 is small, the two indifference curves will be nearly parallel between p 0 and p 1 (indifference curves cannot intersect). In this case, WTP and WTA will be nearly identical. If the risk reduction is large relative to the curvature of the indifference curves, WTA and WTP may be substantially different, with WTA > WTP. 9 For large changes in mortality risk, an individual s WTA compensation in place of an increase in survival probability may be much larger than his or her WTP for the same survival gain (note that WTP is limited by ability to pay, but WTA is not). In principle, the choice of whether WTP or WTA is the appropriate measure of a change in risk may depend on the property right in the situation. If the individual having wealth w 0 is entitled only to the inferior health profile H 0, then it may be appropriate to compare his or her WTP for the improvement to H 1 with the costs of providing the improvement. Alternatively, if the individual is entitled to H 1, then it may be appropriate to compare his or her WTA to forego the improvement with the costs that can be saved by not providing H 1. At the social level, when the costs of reducing risk are born by the beneficiaries, this distinction breaks down. If starting at H 0, the question is whether the individuals collective WTP for an improvement exceeds the cost of improvement and, if 9 If the indifference curve is smooth (which is the case if there are no satiation levels or other thresholds in the individual s preferences for survival probability and wealth), then WTP and WTA for infinitesimal changes in risk are equal. Hanemann (20) shows that indifference curves for a publicly provided good may be curved sharply when no private goods provide close substitutes. In this case, WTP and WTA may diverge substantially even for small changes in the quantity of the public good. The intuition is that the quantity of the publicly provided good (e.g., mortality risk from ambient air pollution) is not subject to the individual s choice. If some private good provides a close substitute, the individual can adjust for a suboptimal quantity of the public good by purchasing more or less of the private good. Thus, if health risk from indoor air quality at home is a close substitute for health risk from ambient air quality, the individual may be able to compensate for poor ambient air quality by investing in cleaner air at home (or for excessively clean ambient air by spending less on controlling indoor pollution). starting at H 1, whether their collective WTA compensation for an increase in risk is less than the costs saved by allowing the increase. The situation in which individuals are entitled to H 1 without paying for it is not logically available in this case. (21) In practice, separate estimates of WTP and WTA can be most easily obtained using contingent valuation or other approaches in which respondents are questioned about their choices in hypothetical situations (described in Section 4). In these cases, estimated WTA is often much larger than estimated WTP. Estimated WTA is often viewed as implausibly large, and so attention has focused on estimating WTP even when WTA might be conceptually more appropriate. (21,22) 3. VALUING MORTALITY RISK In many cases, the health effect of greatest concern is fatality. The effects of individual characteristics, including age, health, competing mortality risk, and income, on the value of reducing mortality risk differ systematically between QALY and WTP approaches. In this section, the effects of these characteristics on the value of reducing a specific current risk (defined as a probability of dying within the current period from a specified cause) are examined under each framework QALYs The value of a change in a specific current mortality risk under the QALY approach is the change in the expected number of QALYs. It depends on life expectancy and expected future health state, but not (with limited exceptions described below) on income or other factors. If the probability of dying from a specific cause in the current period is p, the individual faces a lottery with a p chance of dying in the current period, and a complementary chance of surviving the specific risk and facing the lottery over health profiles that is determined by all the other health risks he or she faces in the current and future periods. Assuming the current period is one year, the health profile if the individual dies from the specific risk provides approximately one-half QALY (assuming he or she is equally likely to die at any time during the year and that his or her HRQL until then is nearly one). The value of a small reduction in the specific fatality risk is: V = pe(qaly) p/2 (3)

7 QALYs Versus WTP 991 where p is the change in the specific risk and E(QALY) is the expected number of QALYs if he or she survives the specific risk. Assuming the expected future QALYs are large compared with 1/2, the second term in Equation (3) can be neglected, yielding: V pe(qaly). (4) As shown by Equation (4), the value of reducing a specific mortality risk depends on the health lottery the individual faces if he or she survives that risk. Indeed, it is nearly proportional to the expected number of QALYs the individual will live if he or she survives. This implies that the value of reducing the specific mortality risk is directly related to the individual s life expectancy conditional on surviving the specific risk, and to his or her expected future health state. For an individual who is likely to survive in very good health (q 1), the value of reducing the specific mortality risk is proportional to life expectancy. For example, the conditional life expectancy of U.S. residents is about 58 years at age 20 and 18 years at age 65, and so the value of reducing a near-term mortality risk to a 20 year old is approximately 3.2 (=58/18) times as large as the value of a comparable risk reduction to a 65 year old. If future QALYs are discounted, the effect of life expectancy is attenuated. Using a recommended discount rate of 3% per annum, (2) the relative value of reducing risk to a 20 year old would be about twice as large as the value of reducing risk to a 65 year old. The effect of a competing mortality risk is to reduce the value of mitigating the specific mortality risk in direct proportion to the magnitude of the competing risk. This follows because the competing risk reduces the expected QALYs conditional on surviving the specific risk. For example, it has been suggested that the individuals who are at greatest risk of dying because of particulate air pollution face very large competing risks because of their age and cardiopulmonary impairments. (23) The associated competing mortality risk may approach one per year, consistent with a life expectancy of less than one year. 10 Under the QALY approach, the value of reducing the risk that such people will die from air pollution is relatively small because their life expectancy conditional on surviving the air pollution is small. 10 Life expectancy of people who die from air pollution is estimated to be on the order of months to years, (24) although for those with chronic obstructive pulmonary disease, the mortality displacement may be on the order of weeks to months. (25) The value of reducing a specific mortality risk is also proportional to the individual s expected future health. Hence, the QALY approach implies it is more valuable to reduce a current mortality risk for someone whose survival would be in very good health than for someone whose survival would be in impaired health. For example, the HRQL for life after a myocardial infarction has been estimated as about 0.9. (26) Under the QALY approach, the value of reducing current mortality risk to someone who has survived a myocardial infarction is about 90% as large as the value of an identical risk reduction to someone who will survive with the same life expectancy but with no significant health impairment. Similarly, if people at risk of death from air pollution have low HRQL because of preexisting illness, the QALY value of reducing mortality risk from air pollution may be lower than the value of reducing risks to healthier people. The relative value of reducing mortality risks to different individuals under the QALY approach is generally considered to be independent of individual economic circumstances, because life years (adjusted for health status) are counted equally regardless of personal characteristics. However, this claim must be qualified, as wealth can have several effects on HRQL, which represents the rate of substitution between longevity and health quality. First, HRQL may depend on individual characteristics and circumstances. For example, the utility consequence of a health impairment may depend on the individual s ability to mitigate it, which may depend on economic circumstances. If the effects of an adverse health condition on individual well-being can be substantially offset using market goods (e.g., personal assistants or mechanical devices), an individual s well-being in that state may be positively related to his or her wealth or income. However, since HRQL measures utility in the impaired health state relative to utility in perfect health, the effect of wealth on HRQL will depend on the relative degree to which it improves well-being in the two states. Under the assumption that QALYs are a utility function for health and longevity, the incremental effect of wealth on welfare is positively associated with health and longevity, except in the implausible case in which incremental wealth is more valuable as a bequest than in life. (27) Limited empirical evidence also suggests that the marginal utility of wealth is smaller in impaired health states than in full health Sloan et al. (28) estimate that having multiple sclerosis (MS) reduces the marginal utility of income by a factor of 0.67 (estimated

8 992 Hammitt Second, under the approach recommended by an expert panel, (2) the effects of health status on earnings capability and income are incorporated in HRQL. 12 The effect of a health impairment on income is likely to depend on both income and the individual s job. Individuals whose income is more sensitive to health status may have a smaller HRQL for the same health impairment (e.g., a physical disability might cause a greater income loss to a construction worker than to a writer). For evaluating the social value of changes in health risk, the effects of income or other individual characteristics on HRQL can be eliminated by valuing all changes using population-average values of HRQL. Indeed, this is the recommended practice. (2) However, if HRQL depends on income, this approach does not aggregate individual changes in welfare and so may lead to ranking health interventions in an order different than the affected individuals would rank them. Note that the same approach using population-average values can be (and usually is) used to remove the effect of income differences on WTP. Since QALYs depend only on the duration and severity of health effects, the value of a risk reduction is independent of other characteristics of the risk, such as whether it is perceived as controllable or dreaded. In principle, the HRQL associated with a health state might be allowed to depend on these characteristics, but this extension has not been investigated WTP Under the WTP approach, the value of reducing mortality risk is measured as the value of a statistical life (VSL). VSL is an individual-specific value defined as the marginal rate of substitution between mortality risk and wealth or income, that is, the individual s WTP for a small reduction in mortality risk divided by the risk change, which is equivalent to the WTA for a small increase in mortality risk divided by the risk change and to the slope of the indifference curve illustrated in Fig. 2 at the individual s wealth and risk level. for people with MS) or by a factor of 0.08 (estimated for people without MS). Similarly, Viscusi and Evans (29) estimate that a workplace accident (which might be fatal or nonfatal) reduces the marginal utility of income by a factor of 0.78 or 0.93 (using alternative functional forms). 12 Brouwer et al. (30) criticize the Gold et al. (2) recommendation and argue that HRQL should be defined to measure preferences for health alone, holding income constant. VSL depends on wealth, current mortality risk, and the lottery over future health profiles the individual faces. The standard model (31 33) assumes that the individual maximizes his or her expected utility: EU(p, w) = (1 p) u a (w) + pu d (w) (5) where p is the individual s chance of dying during the current period and u a (w) and u d (w) represent his or her utility as a function of wealth conditional on surviving and not surviving the period, respectively. The function u d ( ) incorporates the individual s preferences for bequests and can incorporate any financial consequences of dying (such as medical bills or life insurance benefits). In this single-period model, wealth and income are treated as equivalent. In multiperiod models, the difference between wealth and income and the opportunities for future earnings can be important. The individual s VSL is derived by differentiating Equation (5), holding utility constant, to obtain: VSL = dw dp = EU=k u a (w) u d (w) U (1 p) u a (w) + = pu d (w) EU (6) where prime indicates first derivative. The numerator in Equation (6) is the difference in utility between surviving and dying in the current period. The denominator is the expected marginal utility of wealth, that is, the incremental utility associated with additional wealth conditional on surviving and dying in the current period, weighted by the respective probabilities. Assuming that survival is preferred to death (i.e., u a (w) > u d (w)) and that greater wealth is preferred to less (i.e., u a (w) > 0, u d (w) 0), both numerator and denominator are positive and so VSL is positive and the indifference curves in Fig. 2 slope downward. Under the WTP approach, as under the QALY approach, the value of reducing a specific mortality risk in the current period depends on life expectancy, competing mortality risk, and the individual s health if he or she survives the specific risk. In addition, the value under the WTP approach also depends on baseline risk and on income or wealth. First, consider the effect of baseline (total) risk on VSL. It is natural to assume that u a (w) > u d (w), that is, the increased utility provided by greater wealth is larger if the individual survives and has the opportunity to spend it. If so, an increase in the baseline risk p decreases the expected utility cost of spending (the denominator in Equation (6)). The utility associated

9 QALYs Versus WTP 993 with survival (the numerator in Equation (6)) is unaffected by baseline risk, so the individual would be willing to spend more to reduce his or her mortality risk. For small changes in risk, this dead-anyway effect (34) is small. Assuming that u d 0 (i.e., the individual prefers more wealth to less, even if he or she dies), the proportional effect on VSL of a change in baseline risk is less than the proportional change in the survival probability (1 p). The value of reducing the specific mortality risk is smaller if the individual also faces a competing mortality risk. The existence of a competing mortality risk reduces the magnitudes of both the numerator and the denominator in Equation (6). The numerator decreases because the total probability of survival is smaller, and the denominator decreases because of the dead-anyway effect. It can be shown, however, that the effect in the numerator dominates, and so competing mortality risk reduces WTP to reduce the specific mortality risk. (35) VSL may depend on the individual s future health if he or she survives the specific mortality risk, but the sign of the effect is ambiguous. Survival in good health rather than poor increases the value of the numerator in Equation (6). However, if the marginal utility of wealth is higher in good health than in poor health, the value of the denominator is larger and the effect on the ratio is indeterminant. As noted above, limited empirical evidence suggests that the marginal utility of income is smaller in a state of chronic health impairment, (28,29) and some empirical studies suggest that VSL is larger for people with cancer (36,37) or angina (37) than for people without those impairments. As with most goods, WTP for reduction in mortality risk depends on ability to pay and is likely to increase with wealth. The assumption that additional wealth is more valuable in life than as a bequest (i.e., u a (w) > u d (w)) implies that the numerator of Equation (6) increases with wealth. Individuals are generally averse to financial risk. If so, the denominator declines with wealth (the second derivatives of u a (w) and u d (w) are negative), and VSL increases. If the individual is indifferent to financial risk, the denominator is constant and again VSL increases with wealth. Only in the implausible case in which the individual prefers to bear greater financial risk (for the same expected return) can the denominator increase with wealth, making the effect on VSL indeterminate Positive effects of baseline risk and wealth on VSL are sufficient conditions for the convexity of the indifference curves in Fig. 2. The effect of life expectancy on VSL is influenced by two competing factors. A greater life expectancy increases the utility of surviving the current period (the numerator in Equation (6)). Greater life expectancy may also increase the denominator, because of the desire to save wealth for consumption in future periods, or because the opportunity cost of current spending to reduce mortality risk is larger for individuals with a longer investment horizon. The effect of life expectancy also depends on whether the individual is able to borrow against future income and on any difference between the rate at which he or she discounts future utility and the rate of return to savings. A number of investigators have developed theoretical models to examine how VSL varies over an individual s life cycle. These models extend the oneperiod model described in Equation (5) by assuming the individual seeks to maximize the expected discounted value of the utility of consumption: EU = p t δ t u (c t ) (7) t=0 where p t is the probability of surviving at least to age t, c t is consumption at age t, and δ is the individual s discount factor (i.e., δ = 1/(1 + r) where r is the rate at which the individual discounts future utility). In models that assume an individual can borrow against future earnings, VSL declines monotonically with age. Under this assumption, Shepard and Zeckhauser (38) calculate that VSL for a typical American worker falls by a factor of three from age 25 to age 75. If individuals can save but not borrow, VSL rises in early years as the individual s savings (and earnings) increase before it ultimately declines. In this case, Shepard and Zeckhauser (38) calculate that VSL peaks near age 40 and is less than half as large at ages 20 and 65. Ng (39) suggests that individuals may discount their future utility at a rate smaller than the rate of return to financial assets, whereas Shepard and Zeckhauser (38) assume these rates are equal. If the utility-discount rate is less than the rate of return, individuals should save more when they are young and consume more when old. Under these conditions, VSL may not peak until age 60 or so. (39) Even if they discount future utility at the rate of return, prudent (40) individuals might be anticipated to save more and spend less on reducing mortality risk when they are young because of the greater range of financial contingencies they face. WTP may depend on characteristics of the risk other than the probabilities and possible health

10 994 Hammitt outcomes. Limited empirical evidence suggests that average WTP to reduce fatality risks may be somewhat larger for risks that are perceived as involuntary, uncontrollable, unfamiliar, or dreaded. (41 44) For evaluating social programs, it is possible to ignore the effects of individual differences in wealth or other factors that are considered ethically inappropriate by replacing individual VSLs with a value that is obtained by averaging over the objectionable characteristics. This approach is often taken in practice, where differences in wealth and health quality are generally ignored. An alternative approach is to consider how individuals might choose to incorporate differences in wealth and other factors in allocation of social resources if they were to make the decision behind a Rawlsian veil of ignorance before they knew their own characteristics. Pratt and Zeckhauser (34) use this approach to argue that the appropriate VSL for use in social policy choices increases with income, although at a smaller rate than empirical estimates suggest. They also argue that differences in VSL due to differences in baseline risk (the dead-anyway effect) should not be incorporated. 4. METHODS FOR ESTIMATING VALUES Information about preferences, in the form of HRQL or WTP, can in principle be obtained using stated-preference or revealed-preference methods. Stated-preference methods have been used to estimate both HRQL and WTP, but to date revealedpreference methods seem to have been used only to estimate WTP. Stated-preference methods rely on asking individuals either to report their preferences directly or to report how they would behave in a specified hypothetical situation. They are extremely flexible, as individuals can be questioned about how they would choose in a great variety of hypothetical situations. The hypothetical nature of the choice is also the greatest weakness of these methods, as individuals may be unfamiliar with the choices and have inadequate incentive or opportunity to provide thoughtful answers. Revealed-preference methods rely on observing behavior in situations that are more consequential than answering survey questions. They assume that people act in their own best interest and thus the chosen alternative must be preferred to the rejected alternatives. In revealed-preference studies, subjects have an incentive, and may have the opportunity, to seek information about the alternatives and to consider the choice carefully. Nevertheless, individuals may be poorly informed about the differential health risks associated with the choices they face. Also, although the analyst observes the alternatives that individuals choose, he or she does not observe the alternatives they reject and the attributes of those alternatives. This section provides a brief overview of the methods used to estimate HRQL and WTP. The identification of possible health states and probability distribution over time spent in each state that is required for calculating QALYs can be developed using risk assessment methods QALYs HRQL is typically elicited directly or calculated from a generic health utility scale. The generic scales are themselves calibrated using direct elicitation Direct Elicitation The HRQL may be elicited from individuals directly, using any of several question formats: standard gamble, time tradeoff, visual analog scale, and person tradeoff. In general, HRQL for a health state is elicited assuming the health state will be chronic (constant). The standard gamble (SG) format requires the respondent to indicate the smallest chance of survival in perfect health he or she would accept in a lottery where the alternative outcome is immediate death. This may be motivated by considering a surgery that would alleviate a health impairment without affecting longevity, except for the chance of dying in surgery. For example, if the respondent is indifferent between living 20 more years in a particular impaired health state and a lottery that offers her a 75% chance of living 20 more years in perfect health and a complementary chance of immediate death, the value of q for the impaired health state is 3/4, and both the certain health profile and the lottery offer an expected value of 15 QALYs. The time tradeoff (TTO) format requires the respondent to indicate the number of years in perfect health (with q = 1) he or she considers to be indifferent to a specified chronic health profile. For example, if the respondent indicates that she is indifferent between living 20 years in a particular impaired health state and 15 years in perfect health, the value of q for the impaired health state is calculated as 15/20 = 3/4. Both health profiles offer 15 QALYs. The visual analog scale (VAS) is a linear scale with one end representing perfect health and the other

11 QALYs Versus WTP 995 representing health states as bad as death. The respondent is asked to place a mark on the scale representing how desirable the specified health state is to him or her, relative to the endpoints. A similar verbal format may be used where the respondent is asked to report a number representing his or her preference for the health state between 0 and 100, where 0 represents a state as bad as death and 100 represents perfect health. The person tradeoff (PTO) format asks the respondent to consider the relative value of improving health for people in different health states. For example, she might be asked to judge the relative value of extending longevity for people in different health states, e.g., if one were to choose between extending the life of 1,000 healthy people for a year and extending the life of x blind people for a year, for what value of x would she be indifferent? The HRQL of living with blindness is estimated as 1,000/x. Alternatively, the respondent might be asked to judge the relative value of improving health for people in one state and extending life for people in another state, for example, if one were to choose between extending the life of 1,000 healthy people for a year and restoring the site of z blind people for a year, for what value of z would she be indifferent? In this case, the HRQL of living with blindness is estimated as 1 (1,000/z). (18) Risk-tradeoff questions have been used to evaluate preferences for environmental and motor-vehicle related risks. (45 48) In a risk-tradeoff question, respondents are asked to choose between situations offering higher risks of one health outcome (e.g., chronic bronchitis) and lower risks of another (e.g., motor-vehicle fatality). The risk-tradeoff approach is similar to SG. A respondent who is indifferent between reducing his motor-vehicle-fatality risk by 3 per 10,000 and his risk of chronic bronchitis by 1 per 1,000 can be interpreted as having an HRQL for chronic bronchitis of 0.7. If the conditions under which QALYs provide a valid utility function are satisfied, (4 6) and an individual s answers to elicitation questions are consistent with his or her utility function, then both SG and TTO formats should yield exactly the same value. 14 The claim that SG incorporates risk preferences whereas TTO only captures preferences about risk-free outcomes is incorrect. (3) In practice, the results of SG and TTO elicitations differ, perhaps because individuals preferences are not exactly consistent with the required conditions and because the formats make 14 Risk-adjusted QALYs may be written in a form where the answers to SG and TTO questions are not equal but are related to each other by a known transformation. different aspects of the health profiles more salient: SG emphasizes risk and uncertainty, while TTO emphasizes relative preferences for near-term and future health. In practice, SG values may be slightly larger than TTO values. (49) VAS values, because they are not tied to an explicit decision, have a weaker utilitytheoretic justification. In practice, however, they may be more reliably assessed (i.e., vary less on repeat measurement) than TTO or SG values. VAS values are typically smaller than TTO or SG values, but are sometimes adjusted using an empirically estimated formula to approximate the results of TTO or SG formats. Unlike the other methods, PTO has the potential to incorporate judgments about distributional equity. PTO measures preferences over other people s health, and so values elicited using PTO need not correspond to values of HRQL that represent an individual s preferences for his or her own health. An important question in eliciting HRQL is whose values to elicit? Possible respondents include those randomly sampled from the general public, individuals experiencing the health states of interest, and health-care providers or others knowledgeable about the health state. Experience suggests that individuals in an impaired health state assign a larger HRQL to that state than do healthier individuals. Whether this reflects improved understanding of the condition by people experiencing it or adaptation to adverse circumstances is not clear. All the choice-based elicitation methods require comparing two health states, at least one of which is not currently experienced by the respondent at the time of elicitation Generic Health Utility Scales A number of generic health utility scales have been developed. These scales can be used to describe health states in terms of their levels on several attributes, and the HRQL associated with the state may be obtained from a table or calculated using an arithmetic formula. In principle, all such scales are examples of multi-attribute utility functions, although the extent to which the scales are explicitly based on multi-attribute utility theory varies. The scales have been calibrated by fitting them to preference values elicited using one or more of the direct methods reviewed above. Among the more popular generic health utility scales are the Health Utilities Index, (50) the EuroQol EQ-5D, (51) and the Quality of Well-Being Index. (52) The Health Utilities Index Mark 3 (HUI3) classifies health states using a system of eight attributes: